exponentiation

Examples of exponentiation in the following topics:

• Exponentials With Complex Arguments: Euler's Formula

• Graphs of Exponential Functions, Base e

  • The function $f(x) = e^x$ is a basic exponential function with some very interesting properties.
  • The basic exponential function, sometimes referred to as the exponential function, is $f(x)=e^{x}$ where $e$ is the number (approximately 2.718281828) described previously.
  • The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable.
  • If the change is positive, this is called exponential growth and if it is negative, it is called exponential decay.
  • For example, because a radioactive substance decays at a rate proportional to the amount of the substance present, the amount of the substance present at a given time can be modeled with an exponential function.

• Simplifying Exponential Expressions

  • Multiplying exponential expressions with the same base ($a^m \cdot a^n = a^{m+n}$)
  • In terms of conducting operations, exponential expressions that contain variables are treated just as though they are composed of integers.
  • Applying the rule for dividing exponential expressions with the same base, we have:
  • To simplify the first part of the expression, apply the rule for multiplying two exponential expressions with the same base:

• Exponential Decay

  • Exponential decay is the result of a function that decreases in proportion to its current value.
  • Just as it is possible for a variable to grow exponentially as a function of another, so can the a variable decrease exponentially.
  • Exponential rate of change can be modeled algebraically by the following formula:
  • The exponential decay of the substance is a time-dependent decline and a prime example of exponential decay.
  • Below is a graph highlighting exponential decay of a radioactive substance.

• Converting between Exponential and Logarithmic Equations

  • Logarithmic equations can be written as exponential equations and vice versa.
  • The logarithmic equation $\log_b(x)=c$ corresponds to the exponential equation $b^{c}=x$.
  • It might be more familiar if we convert the equation to exponential form giving us:
  • If we write the logarithmic equation as an exponential equation we obtain:
  • We can use this fact to solve such exponential equations as follows:

• Introduction to Exponential and Logarithmic Functions

  • Logarithmic functions and exponential functions are inverses of each other.
  • The inverse of an exponential function is a logarithmic function and vice versa.
  • In the following graph you can see an exponential function in red and its inverse, a logarithmic function, in blue.
  • The natural logarithm is the inverse of the exponential function $f(x)=e^x$.

• Basics of Graphing Exponential Functions

  • The most basic exponential function is a function of the form $y=b^x$ where $b$ is a positive number.
  • This is called exponential growth.
  • This is called exponential decay.
  • This is true of the graph of all exponential functions of the form $y=b^x$ for $x>1$.
  • This is true of the graph of all exponential functions of the form $y=b^x$ for $0

• Introduction to Exponents

  • Exponential form, written $b^n$, represents multiplying the base $b$ times itself $n$ times.
  • Exponentiation is a mathematical operation that represents repeated multiplication.
  • Now that we understand the basic idea, let's practice simplifying some exponential expressions.
  • Let's look at an exponential expression with 2 as the base and 3 as the exponent:
  • Let's look at another exponential expression, this time with 3 as the base and 5 as the exponent:

• Problem-Solving

  • The exponential function has numerous applications.
  • Economic growth is expressed in percentage terms, implying exponential growth.
  • Compound interest at a constant interest rate provides exponential growth of the capital.
  • Graph of an exponential function with the equation $y=2^x$.
  • In what other circumstances would you see exponential growth?

• Logarithmic Functions

  • Logarithmic equations can be written as exponential equations and vice versa.
  • The logarithmic equation $log_b(x)=c$ corresponds to the exponential equation $b^{c}=x$.
  • As an example, the logarithmic equation $log{_2}16=4$ corresponds to the exponential equation $2^4=16$.
  • It might be more familiar if we convert the equation to exponential form giving us:
  • The explanation of the previous example reveals the inverse of the logarithmic operation: exponentiation.